Abstract
Two systems of Earth-centered inertial Newtonian orbital equations for a spherical Earth and three systems of post-Newtonian nonlinear equations, derived from the second post-Newtonian approximation to the Earth Schwarzschild field, are used to carry out a performance analysis of a numerical procedure based on the Dormand-Prince method for initial value problems in ordinary differential equations. This procedure provides preliminary post-Newtonian corrections to the Newtonian trajectories of middle-size space objects with respect to space-based acquisition, pointing, and tracking laser systems, and it turns out to be highly efficient. In fact, we can show that running the standard adaptive ode45 MATLAB routine with the absolute and relative tolerance, TOLa = 10−16 and TOLr = 10−13, respectively, provides corrections that are final within the eclipses caused by the Earth and close to final during the noneclipse phases. These corrections should be taken into account to increase the pointing accuracy in implementing the space-to-space laser links required for ablation of designated objects or communications between space terminals.
Highlights
In a previous paper [1], the main post-Newtonian (p-N) corrections to the Newtonian trajectories of middle-size LEO space debris object D with respect to acquisition, pointing, and tracking (APT) laser system S, aimed at ablating D, were derived by means of a genuine p-N method for the Earth Schwarzschild field [2]
After the Earth-centered inertial (ECI) orbital coordinates of S and D are determined from appropriated p-N orbital equations, the main difficulty of this method is that the process to track D involves computations of the time-dependent coefficients of a third system of p-N equations for the relative motion of D with respect to S
In order to derive the results below, the ECI orbital motions of S and D are assumed to be equatorial, so that the number of equations which we need to solve for the p-N orbit of D with respect to S reduces from eighteen to twelve
Summary
In a previous paper [1], the main post-Newtonian (p-N) corrections to the Newtonian trajectories of middle-size LEO space debris object D with respect to acquisition, pointing, and tracking (APT) laser system S, aimed at ablating D, were derived by means of a genuine p-N method for the Earth Schwarzschild field [2]. After the Earth-centered inertial (ECI) orbital coordinates of S and D are determined from appropriated p-N orbital equations, the main difficulty of this method is that the process to track D involves computations of the time-dependent coefficients of a third system of p-N equations for the relative motion of D with respect to S These coefficients are line integrals of the Riemann tensor of space along the line of sight (LOS) segment that joins S and D (cf [1]). We are entitled to adopt this rule, since the Riemann parallel transport in this approximation is the second p-N approximation to the Euclidean transport In those scenarios where eclipses may occur, one has to consider the two expected alternatives, eclipse/ noneclipse, by means of the “if/else” computational command, and make the respective p-N relative trajectories match. We compute the values of the corrections by means of the TER, which can be considered as final within the eclipse phases and as preliminary for the noneclipse phases
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